We report the formation of stable bound wave packets in a modified Belousov-Zhabotinsky reaction. These densely stacked structures arise from an attractive interaction between oxidation pulses that is not known from the classical Belousov-Zhabotinsky system. The characteristic stacking period increases with the initial concentration of bromate but decreases with cyclohexanedione. Wave stacking can also induce cascades of bunching events in which internally dense but mutually well segregated wave clusters are formed. For different initial concentrations, we observed the apparent merging of waves in front-to-back collisions. All three modes of wave dynamics are analyzed in terms of their dispersion behavior. The dispersion relations proved to be anomalous in each case and revealed the existence of an attractor which induces the formation of stable wave packets. The underlying mechanism has a pure reaction-diffusion character since wave propagation is not affected by fluid convection. At high initial concentrations of ferroin, we detected complex relaxation kinetics which indicate the presence of at least two independent species that control the recovery and hence the dispersion behavior of the medium.